# Questions tagged [copula]

A copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the unit interval.

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### Calculation on Quadrant Probability for Bivariate data using Copula

I have a question regarding the computation on bivariate probability when using copula function. Let $u=F_X(x)$ and $v=F_T(t)$ be the CDF for marginals of $X$ and $T$ respectively, a joint ...
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### Implied Correlation from Gaussian Copula

I am building a spreadsheet model that allows marginal distributions to be correlated together using a Gaussian copula (with prescribed correlation matrix). The inputs into the model are the ...
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### Indefinite integral | How to compute the second-order partial derivatives of Mixed (2 gaussian & 2 binary) Gaussian Copula?

This is a problem about "Computing the Mixture Gaussian Copula with 2 normal (continuous) variables and 2 binary (discrete) variables". Problem background I have two continuous r.v. and two ...
1 vote
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### How is the copula $C\big(X_1, 1/X_1, \exp(-X_1)\big)$ equal to the copula $C(X_1, - X_1, -X_1)$ and what is its form?

Let $X_1$ be a positive random variable with a continuous cumulative distribution function and $C$ the copula of $(X_1, 1/X_1, \exp(-X_1))$. Why is $C$ also the copula of $(X_1, -X_1, -X_1)$ and what ...
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### How to prove that the multivariate Frechet-Hoeffding lower bound is not copula when $n \geq 3$?

I want to show that the multivariate Frechet-Hoeffding lower bound given by $C^-(u_1,\dots, u_n) = \left(\sum_{i=1}^{n}{u_i} - (n-1)\right)_+$ is not a copula when $n \geq 3$. My question is: Is it ...
55 views

### Semi-survival copula

I'm currently doing some reading on copula function. For a bivariate cdf, by Sklar's theorem, there exists a copula function,C, such that $P(U<u,V<v) = C(u, v)$. As written by Nelsen (2006), A ...
so i got the following problem: consider the follow. $$\overline{C}_{\overline{X}}:\mathbb{R}\to [0,1], u\mapsto\overline{C}_{\overline{X}}(u) = \mathbb{P}(\overline{F}_{X_1}(X_1)\leq u_1,...,\... 1 vote 1 answer 33 views ### Multivariate t distribution: Find probability of region enclosed by constant-density hypersurface I am working with a multivariate t distribution, say of dimension p. Given a point P = (x1, ..., xp) in the sample space I need to calculate the probability of the region of the sample space enclosed ... 1 vote 0 answers 34 views ### Monotonicity of Parametric Bivariate Copula w.r.t. \theta Let C(u_1,u_2;\theta) be a bivariate parametric copula. I know if C(u_1,u_2;\theta) is the Gaussian copula, then \partial_{\theta} C(u_1,u_2;\theta)>0 for any u_1 and u_2 (it follows from ... 3 votes 0 answers 48 views ### How do I derive a pair-copula decomposition for a joint density function? In Section 4.1 of Analyzing Dependent Data with Vine Copulas (Czado), the author decomposes a three-dimensional joint density function into bivariate copula densities and marginal density functions. I’... 1 vote 1 answer 52 views ### Why can this joint distribution function be written as this integral of a conditional distribution function? In Section 3.8 of Dependence Modeling with Copulas (Joe), the author starts with the following. In this section, we show how Sklar's theorem applies to a set of univariate conditional distributions, ... 1 vote 1 answer 26 views ### Finding a value \alpha such that a function c is a copula density of (U_1, U_2) Let$$c(u_1,u_2) = 1 + \alpha(1- 2u_1)(2- 2u_2)$$where u_1, u_2 \in (0,1). The question is, for which \alpha is the function c a copula density for (U_1, U_2). So my idea was to compute the ... 0 votes 0 answers 25 views ### Beyond vine copulas I know that vine copulas allow for different dependency models between pairs of variables. I wondered if they are also used to model dependency models between subsets (beyond pairs) of variables or if ... 1 vote 1 answer 100 views ### Sum of dependent random variables and copulas I have two dependent continuous random variables (RVs) X and Y and I'm interested in determining the CDF of the sum, i.e., F_{X+Y}(t) = \mathbb{P}(X+Y \leq t). I know the marginal of X and Y ... 4 votes 1 answer 145 views ### Proving that copulas are Lipschitz continuous A copula is a function C:[0,1]^2\to[0,1] such that C(x,0)=C(0,x)=0 for all x\in[0,1], C(x,1)=C(1,x)=x for all x\in[0,1], and \begin{equation}\label{ineq} C(x_2,y_2)-C(x_1,y_2)-C(x_2,y_1)+C(... 0 votes 1 answer 27 views ### Calculating the joint cumulative distribution function from a junction tree Assume I have the following Junction Tree between random variables X_1,\dots,X_7 that exactly describes the sets variables with non-zero Mutual Information (Alternatively it's the last tree in a ... 5 votes 0 answers 219 views ### Product of correlated random variables and its transformation There is an interesting result, saying that if Z_1, Z_2 are standard normal random variables with a correlation \rho\in (-1,1), then the product Z=Z_1Z_2 has a density function explicitly given ... 1 vote 0 answers 45 views ### Combining conditional probabilities using an unconditional copula First: just a bit of background on copulas. Suppose we have a pair of continuous random variables Y_1, Y_2 with distribution functions F_1(y_1)=P(Y_1\leq y_1) and F_2(y_2)=P(Y_2\leq y_2). Let ... 2 votes 1 answer 153 views ### Proving that C(x_1,x_2)=\max\{x_1+x_2-1,0\} is a copula I have the following problem. I need to prove the copula property for the function C(x_1,x_2)=\max\{x_1+x_2-1,0\}, better known as the lower Fréchet-Hoeffding-Boundary. A copula is defined as a ... 1 vote 0 answers 21 views ### Frequently used methods to estimate copula of discrete random variable in high dimension I am new to the estimation of copula with discrete marginal distributions. For example, I have the data of 30 discrete random count variables from X_1 to X_{30}, some X_j has Poisson ... 1 vote 1 answer 183 views ### Kendall’s tau between X and 1/Y I'm currently studying for my statistics exam by doing exercises from the book Statistics and Data Analysis for Financial Engineering with R examples. I'm struggling with this exercise from chapter 8: ... 1 vote 1 answer 119 views ### Proof that Pearson correlation is/isn't a functional of the copula for a given pair of random variables I have a growing interest and respect for the subject of copulas initially thanks to comments made on stats.SE by kjetil-b-halvorsen. The most interesting to me right now is the following: "[T]... 1 vote 0 answers 279 views ### Sampling from Gaussian Copula with a conditional distribution approach In the paper "Cheng and al. (2007)" on pages 193-194, the authors propose an algorithm that generates variables with a given copula function C being the joint distribution. This procedure ... 2 votes 1 answer 326 views ### Showing that the lower bivariate Fréchet-Hoeffding bound is a copula. I want to show that the bivariate Fréchet-Hoefdding lower bound is indeed a copula. The bivariate function is defined by: W(x,y)=\max\{x+y-1,0\}, \ x,y\in[0,1]  Definition "Copula": A two-... 1 vote 1 answer 340 views ### How to deal with copulas? I want to model a couple of variables (X,Y) using copulas. My idea is to model the marginal distributions and then use copulas to combine marginal distributions and copula to get a joint ... 1 vote 0 answers 50 views ### Kendall's tau for archimedian copula Consider a 2-dimensional archimedian copula with generator \psi and \gamma:=\psi^{-1}, i.e.$$C(u,v)=\gamma(\psi(u)+\psi(v))$$I want to show that$$\tau=1-\int_{0}^{\infty}t\gamma'(t)^2dt$$I ... 1 vote 0 answers 25 views ### Find \lim_{x\uparrow 1}\mathbb{P}(F_Y(Y)>x|F_X(X)>x) where X,Y poisson Let X=Y_1+N,Y=Y_2+N where N\sim \text{pois}(\lambda), Y_1\sim \text{pois}(\lambda_1), Y_2\sim \text{pois}(\lambda_2) and N,Y_1,Y_2 are independent. I'm asked to find$$\lim_{x\uparrow 1}\mathbb{... 